Is someone allowed to manually change sign of an estimate (obtained through a OLS), if this is supported by an underlying theory?
My first idea was that this is basically a manipulation, which forces the estimates to go in the direction we want to; in fact, the sign of the estimate should directly capture the "right" direction of the effect. However, I am starting to think that it might be justified in some cases.
My case is the following. My assumption is that a covariate X has a negative effect on Y, however X is represented by a proxy X1 (which is what eventually is included in the model; X does not appear in the regression). So, this is like saying that X is a function of X1. X1 can either have a negative or a positive effect on X. Thus, the sign of the estimate of B1 is negative (as expected since it proxies the effect of a variable with a negative impact on Y), when the impact of X1 on X is also negative, but it is positive when the impact of X1 on X is positive. In the latter situation, can I put a - in front of B1 and justify in the way I did now (yet more formally)?
I should correct myself. We have not changed the sign of the coefficient, but we changed sign of the observed values for this variable. We have experimental data, and we use proxies for competition on the demand side of the labour market (our variable of interest); we know from the previous literature that the competition has a negative effect on the dependent variable. 2 proxies we used are negatively related to Y, and thus also on competition (the proxy decreases in value, the competition increases, Y decreases). The third proxie we use is a rate and moves instead on the same direction of competition (increase in the rate, increase in the competition), therefore it has a positive effect on Y, which makes it a bad proxy. So, I thought we might use the opposite of this variable to be a proxy for competition, so that we keep the negative relationship between the proxy and the dependent variable. The opposite of this rate (by that I mean: 1-rate) is a measure that actually makes sense.